I. Field of the Invention
The present invention relates to Magnetic Resonance Imaging (MRI) and more particularly to methods and related apparatus for removing a phase error in digitized MRI data resulting from, inter alia, quadrature demodulation of an MRI echo signal from an observed object.
II. Background Information
A Two Dimensional Fourier Transform (2DFT) MRI system such as that discussed by A. Kumar, et al. in J. Magn. Reson. 18: 69-83, 1975, subjects an object, such as a human body, to a series of magnetic fields and radio frequency (RF) pulses. The RF pulses resonate selected atomic nuclei in the object as a consequence of the characteristics of the atoms, the applied magnetic fields, and the RF pulses. The resonant nuclei precess in the magnetic fields and radiate a detectable RF echo signal containing information about the location of the radiating atoms. As is well known to those skilled in the art, the echo, when quadrature demodulated by a reference waveform, is a sample of the two dimensional Fourier transform, designated F(X,Y), of the object density per unit area, designated f(x,y), of the selected nuclei in a slice plane through the object defined by the magnetic field gradients. The object density function f(x,y) represents the spatial distribution of the resonant atoms and their relaxation times in the object, and is used to construct a visual image of some property of the object along the slice plane or volume. The object density function is a function of the spin density .rho., spin lattice relaxation time T.sub.1, and the spin-spin relaxation time T.sub.2 at each point x,y in real space on a plane passing through the object under observation. Because of unknown and time varying differences between the phase of the echo signal from the object and the quadrature demodulation reference waveform, the demodulation process used in the collection of complex data from a phase sensitive detector system introduces a phase error into the MRI data that distorts the final visual image.
If the object under observation is a human body, then the spatial coordinates x, y, and z may be oriented with respect to the body as shown in FIG. 1. By subjecting the object to a static magnetic field, usually parallel to the Z axis, and gradient magnetic fields, data values represented by the function F(X,Y) can be collected along specific paths on the Fourier space plane defined by: ##EQU1## The function g.sub.x is the "read" magnetic field gradient with a gradient direction parallel to the x-axis, g.sub.y is the "encode" magnetic field gradient with a gradient parallel to the y-axis, and .gamma. is the magnetogyric ratio of the atoms under observation. Selection of the static and gradient fields defines the location and orientation of the Fourier space coordinates.
One proposed error correction method, described in U.S. Pat. No. 4,706,027, incorporated herein by reference, estimates and corrects the phase error occurring in scan lines which do not pass through the origin of Fourier space. That technique requires an estimate of the phase error in F(X,Y) at the origin of Fourier space, F(0,0), based on the theorem that the Fourier transform F(0,0) is real at the origin because f(0,0) is real. Any complex components of f(0,0) must therefore be the result of a phase error such as that introduced by the quadrature demodulation process. For the theorem to be applied, the value of the one data value at the origin in Fourier space must be known. To find the data value at the origin, a scan line is generated that passes through the origin. The data values comprising any scan line through the origin are called central section data values, and the scan line is called a central section scan line.
In existing MRI scanners, the phase error is calculated based on a zero location and correction calculation which estimates the location of the peak value of .vertline.F(X,Y).vertline. which is assumed to be the value at the origin of Fourier space. Often the peak falls at some location between two data values, as shown in FIG. 4, therefore the location of the peak must be interpolated. The estimate of the phase error therefore depends on an interpolation calculated from three observed data values, each of which is subject to experimental error. Because of noise in the received signals, the calculated value of F(0,0) is not guaranteed to be the actual value at the origin, resulting in additional errors in the phase error calculation.
Another source of errors in the phase error calculation is the decay of the magnitude of RF echo signals, indicated by T.sub.2. The T.sub.2 decay results from slightly different precession rates of the resonant atomic nuclei in the object which are subject to inhomogeneous magnetic fields from the MRI system magnets and from other molecules.